## Busy times, 2

Some progress since the last report: the marking is nearly finished! This is just as well, since they have spent the morning drilling in the courtyard and I can’t stand a lot more of it.

At first it was rather depressing; many students were doing very badly. I had to resist the thought that they had let me down, and I would be leaving teaching at Queen Mary on a low point. In fact, of course, they have only let themselves down. But the first packet I chose to mark seemed to be below average, and things have been a bit better since. Indeed, at the very top, the performances this year seem even better than last year.

So I seem to have scattered the marks over the whole scale. This, of course, is not popular with the bureaucrats today, who want the vast majority of students to score between 40 and 80. Fortunately I don’t have to worry about this. I can’t even enter the marks, since the Queen Mary computers have decided I am a non-person as far as teaching matters go.

I don’t want to give a long list of howlers. But here are some things worth pondering.

• What is the negation of the statement “At least one of the numbers … is divisible by p“? Not so difficult, you might think. But the most common response is “At least one of the numbers … is not divisible by p“; and “At most one of the numbers … is divisible by p” is also quite popular.
• I have mentioned before that, in a proof by induction, students have a very strong tendency that the way to do the induction step is to add the same thing to both sides of some equation. I think this is probably because, in many cases, when they meet induction for the first time, teachers use stock examples such as showing that the sum of the first n squares is given by some formula. I tried to give the students a view of the enormous flexibility of induction as a proof tool, but clearly not all of them were able to put aside their early conditioning.
• A surprising number of people, needing to prove that x+x is even for any integer x, separate into two cases, x even and x odd. Why?
• I made a rod for my own back by having a question in which part (c) said, “Use your method of proof [for part (b)] to do …”. If students did … otherwise, they lost marks. But those whose proof in part (b) was entirely bogus created more of a problem for me, since any argument might be said to follow from a bogus proof.

I did learn one beautiful thing from thinking about some failed attempts to prove that there are infinitely many primes. Several students said, “If p were the largest prime then 2p−1 would be a larger prime, a contradiction”, and justified this by pointing out that 22−1, 23−1 and 25−1 are all primes.

The argument can be fixed like this (though a little more than basic first-year maths is needed). Suppose that p were the largest prime. Consider 2p−1. This may not be prime, but let q be a prime divisor of it. Then 2p is congruent to 1 (mod q), so the order of 2 mod q divides p, and so is equal to p since p is prime. So p divides the order of the multiplicative group of integers mod q, which is q−1. So, necessarily, q is larger than p.

Did I just invent that, or have I seen it before? Maybe it is in Proofs from the Book

By taking small breaks from marking, I have managed to finish the slides at least for the lectures on infinity; a very pleasant diversion!

I count all the things that need to be counted.
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### 1 Response to Busy times, 2

1. Christian Elsholtz says:

This is indeed in “Proofs from THE BOOK”.
Another good source for Euclid-type proofs is
“Euclid’s theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.–2012) and another new proof”, by Romeo Mestrovic, see http://arxiv.org/abs/1202.3670
On page 16 there is this proof, and further references.

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